Guess if nobody else is going to answer......... I think, BD is technically the same but from a different breeder. Unless you live in Calif. in which you have the true BDH clone. But don't worry about your seed if it's not from Cali Connections, if it's from HSO it may be fine.
Blue Dream Haze is a "clone only" plant from CA. The seeds come from the magic of chemistry. Taking a clone, chemically switching it's sex ( sort of as is still a female plant but with male sexual organs), and breeding to another clone. Tricky business as you have really bred to a hermie, and that's never good. The big prob seems to be that Cali rushed to market before they stabilized this strain. Most people lose them to hermie, those very very few who don't have low yields. But killer smoke. Ha.
Anyway, if you have HSO Blue Dream I'd love to hear your results. so far I haven't had enough info on their seed and after being burned by this strain I don't think I want to try a "new" strain for a long time.
Hope this helps.